No! You can only go inside during the Summer Opening, and that always takes place when the Queen is away at Balmoral Castle in Scotland (she holidays there every year).
Other members of the family live elsewhere so you won't see them either.
You just get to tour the elaborately-decorated State Rooms at the back and look round the garden. The Queen and the Duke of Edinburgh actually live in apartments in a side wing (on the right as you look at the palace from the front) and a lot of the rest is offices so it wouldn't be interesting. The biggest room is the Ballroom, used for investitures (presentation of honours) and state banquets, but that's not on the tour either - I know from being on it. It's a fairly plain room anyway, just big.
Just for fun, the most likely time members of the public might actually meet the Queen is at a Garden Party. Even that is by invitation only and there will be some reason why you get invited. And that's a dressing-up occasion - men should wear suits, women should wear suitable day dresses and hats or fascinators, "afternoon tea" is served from tents in the garden, and you MIGHT get a few words with Her Majesty. All very nice unless it rains!